Embedding minors in expanders has received attention in combinatorics [1]} and theoretical computer science, finding applications in proof complexity [2]}.
Kleinberg and Rubinfeld [3]} proved that if \(G=(V,E)\)  is a \(\alpha \) -expander with maximum degree \(\Delta \) , then every graph with \(n/\log ^{\kappa } n\)  vertices and edges is a minor of \(G\)  for a constant \(\kappa >1\)  depending on \(\Delta \)  and \(\alpha \) . Later it was stated [4]} that \(\kappa (\Delta , \alpha )=\Omega (\log ^2(d)/ \log ^2(1/\alpha ))\) . Krivelevich [5]} together with Nandov proved that if \(G\)  is an \(\alpha \) -vertex expander then it contains every graph with \(cn/\log n\)  edges and vertices for some universal constant \(c>0\) .
